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 A087803 Number of unlabeled rooted trees with at most n nodes. 22
 1, 2, 4, 8, 17, 37, 85, 200, 486, 1205, 3047, 7813, 20299, 53272, 141083, 376464, 1011311, 2732470, 7421146, 20247374, 55469206, 152524387, 420807242, 1164532226, 3231706871, 8991343381, 25075077710, 70082143979, 196268698287, 550695545884, 1547867058882 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Number of equations (order conditions) that must be satisfied to achieve order n in the construction of a Runge-Kutta method for the numerical solution of an ordinary differential equation. - Hugo Pfoertner, Oct 12 2003 REFERENCES Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations, (1987) Wiley, Chichester See link for more references. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268. I. Th. Famelis, S. N. Papakostas and Ch. Tsitouras, Symbolic Derivation of Runge-Kutta Order Conditions. R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy). R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876. Eric Weisstein's World of Mathematics, Rooted Tree. FORMULA a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.664861031240097088000569... . - Vaclav Kotesovec, Sep 11 2014 In the asymptotics above the constant c = A187770 / (1 - 1 / A051491). - Vladimir Reshetnikov, Aug 12 2016 MAPLE with(numtheory): b:= proc(n) option remember; local d, j; `if`(n<=1, n,       (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))     end: a:= proc(n) option remember; b(n) +`if`(n<1, 0, a(n-1)) end: seq(a(n), n=1..50);  # Alois P. Heinz, Aug 21 2012 MATHEMATICA b[0] = 0; b[1] = 1; b[n_] := b[n] = Sum[b[n - j]* DivisorSum[j, # *b[#]&], {j, 1, n-1}]/(n-1); a[1] = 1; a[n_] := a[n] = b[n] + a[n-1]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *) t[1] = a[1] = 1; t[n_] := t[n] = Sum[k t[k] t[n - k m]/(n-1), {k, n}, {m, (n-1)/k}]; a[n_] := a[n] = a[n-1] + t[n]; Table[a[n], {n, 40}] (* Vladimir Reshetnikov, Aug 12 2016 *) Needs["NumericalDifferentialEquationAnalysis`"] Drop[Accumulate[Join[{0}, ButcherTreeCount[20]]], 1] (* Peter Luschny, Aug 18 2016 *) PROG (PARI) a000081(k) = local(A = x); if( k<1, 0, for( j=1, k-1, A /= (1 - x^j + x * O(x^k))^polcoeff(A, j)); polcoeff(A, k)); a(n) = sum(k=1, n, a000081(k)) \\ Altug Alkan, Nov 10 2015 (Sage) def A087803_list(len):     a, t = [1], [0, 1]     for n in (1..len-1):         S = [t[n-k+1]*sum(d*t[d] for d in divisors(k)) for k in (1..n)]         t.append(sum(S)//n)         a.append(a[-1]+t[-1])     return a A087803_list(20) # Peter Luschny, Aug 18 2016 CROSSREFS a(n) = Sum_(k=1..n) A000081(k). Cf. A255170, A187770, A051491. Sequence in context: A324936 A003426 A179476 * A212658 A036374 A214999 Adjacent sequences:  A087800 A087801 A087802 * A087804 A087805 A087806 KEYWORD nonn AUTHOR Hugo Pfoertner, Oct 12 2003 EXTENSIONS Corrected and extended by Alois P. Heinz, Aug 21 2012. Renamed (old name is in comments) by Vladimir Reshetnikov, Aug 23 2016. STATUS approved

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Last modified October 16 13:51 EDT 2019. Contains 328093 sequences. (Running on oeis4.)